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International Journal of Academic Research in Business and Social Sciences

Vol. 8 , No. 11, Nov, 2018, E-ISSN: 22 22 -6990 © 2018 HRMARS

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An Estimation of Value at Risk using GARCH Models for the Conventional and Islamic Stock Market in Malaysia

Nor Azliana Aridi, Chin Wen Cheong, Tan Siow Hooi

To Link this Article: http://dx.doi.org/10.6007/IJARBSS/v8-i11/5568 DOI: 10.6007/IJARBSS/v8-i11/5568

Received: 28 Oct 2018, Revised: 25 Nov 2018, Accepted: 30 Nov 2018

Published Online: 04 Dec 2018

In-Text Citation: (Aridi, Cheong, & Hooi, 2018) To Cite this Article: Aridi, N. A., Cheong, C. W., & Hooi, T. S. (2018). An Estimation of Value at Risk using GARCH

Models for the Conventional and Islamic Stock Market in Malaysia. International Journal of Academic Research in Business and Social Sciences, 8(11), 2054–2065.

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An Estimation of Value at Risk using GARCH Models for the Conventional and Islamic Stock Market in

Malaysia

Nor Azliana Aridi1, Chin Wen Cheong2, Tan Siow Hooi3 1Lecturer, Quantitative Method Unit, Faculty of Management, Multimedia University, 63100

Cyberjaya, Malaysia. 2Associate Professor, Department of Mathematics, Xiamen University Malaysia, Jalan Sunsuria,

Bandar Sunsuria, Sepang, 43900, Selangor, Malaysia. 3Associate Professor, Economics Unit, Faculty of Management, Multimedia University, 63100

Cyberjaya, Malaysia.

ABSTRACT This study attempts to estimate the value-at-risk (VaR) to forecast volatility for both conventional and Islamic stock markets in Malaysia. In particular, the purpose of the article is to investigate whether GARCH models are accurate in the evaluation of VaR in emerging stock markets such as Malaysia.The daily return of the conventional (KLCI) and Islamic (FBMS) stock market are analysed for the period 2000 – 2015. The volatility model of GARCH (1,1), TGARCH (1,1) and CGARCH (1,1) with a normal and and student-t distribution are used to model the conditional variance of the stock market returns. The VaR violations of unconditional coverage and the backtesting procedure of Kupiec test are used to check the reliability and accuracy of the volatility model used for both normal and student-t distribution. Based on the Akaike Information Criterion (AIC), the best model for modelling the conventional and Islamic stock market returns is TGARCH (1,1). The backtesting results showed that for all GARCH models used, the normal distribution gives better forecast VaR compared to the student’s t distribution. Keywords: Value at Risk, Heavy Tails, GARCH Model, Volatility forecasting, Backtesting INTRODUCTION The modern portfolio theory (MPT) describes the strong relationship between risk and volatility where volatility generates risk associated with the level of dispersion around the mean. In stock market situation, the greater the standard deviation means the greater the stock market dispersion of returns and the greater the risk of the stock market. In other words, the investment is riskier given that the chance to generate excess returns is higher. Therefore, investors who desire for a higher



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return must also prepare for any financial uncertainty. Many previous studies have found models and theories with sophisticated tools that are applicable in measuring and forecasting the stock market risk. One of the common used risk measure is called value at risk (VaR). VaR is used to calculate the maximum financial loss over a specific time frame for a given confidence level by focusing on the estimation of the tails distribution. The primary reason for the underestimation or overestimation of VaR is because numerous application in the financial field assume that stock market returns are normally distributed while in the real world it is leptokurtosis which exhibit skewness and excess kurtosis. This paper aims to explain the most accurate model to forecast volatility and analyzed the existence of empirical facts in conventional and Islamic market in Malaysia. This paper is organized into seven sections. Section one provides a literature review follows with the methodology used in Section three. Section four discusses the backtesting procedure to check the accuracy of the model used. Next, section five and six provides the descriptive statistics and empirical findings. Finally, section seven concludes the paper. According to (Goudarzi , 2011), it is essential to continuous study and find out the most accurate measurement of volatility in assets pricing, risk and portfolio management. The application that commonly used in estimating the volatility of stock market returns is the ARCH (GARCH) models. This is due to the capability of these models to capture the characteristics of return series and identify the random movement of stock market prices. Previous studies show no final decision on which one is the best model to forecast volatility. There were studies present a good forecast result based on the simple GARCH (p,q) models but some present that the extensions of GARCH models such as TARCH, EGARCH, APARCH and others provide a better forecast result. In research study done by (Angelidisa, Benosa, & Degiannakis, 2004) on S&P 500 Index proves that the combination of a leptokurtic distribution and a simple asymmetric volatility model AR (1) EGARCH (0, 1), is a good combination in estimating the VaR. Study done by (Smolović, Lipovina-Božović , & Vujošević, 2017) joined the ARMA model with different specifications of conditional heteroscedasticity models and found that GARCH (1,1), APARCH (1,1) and TGARCH (1,1) are accurate to forecast VaR in emerging stock market. Finding from (Li, Lian , & Zhang, 2016) presents that EGARCH (1,1) is the best forecasted model for Europe Brent crude oil while APARCH (1,1) and GJR-GARCH (1,1) outperformed other GARCH models for West Texas Intermediate (WTI) Cushing crude oil. The forecasting VaR procedure can also be used for the purpose of finding whether Islamic or Conventional stock market are less volatile to the financial uncertainty (Abdrashev, 2016). Study done by (Bhatt & Sultan, 2012) shows that the Shariah stock market has lower risk premium than the conventional market. In term of the risk premium, (Albaity & Ahmad, 2011) show that Islamic markets has leverage effects but no risk premium. Another study in Islamic market has been done by (Nasr, Lux , Ajmid, & Guptae, 2016) on the Dow Jones Islamic Market World Index (DJIM) and the result presents that Markov-switching multifractal (MSM) model outperform the GARCH, FIGARCH and FITVGARCH models in the prediction of VaR. METHODOLOGY The data used are the daily return for FTSE Bursa Malaysia KLCI Index (KLCI), FTSE Bursa Malaysia EMAS Shariah Index (FBMS) from year 2000 – 2015. The continuously compounded rate of return is given by:



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1

ln tt

t

Pr

P−

=

(1)

Given that tP is the closing price of the security at time t, the stock market returns contain predictable

and unpredictable returns as presented in equation (2).

1( )t t t tR E R −= + (2)

where 1t − is all information available at time 1t − and t is the unpredictable return written as

t t tz = , t is the time varying volatility and tz is iid with mean zero and standard deviation one,

(0,1)tz N . In estimating the log return series, the ARCH (q) model which was introduced by (Engle,

Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, 1982) was used to represents the conditional mean equation thus the conditional variance is expressed as a linear function of the previous q squared innovations written as:

2 2

1

p

t i t ii −=

= + (3)

To make sure that the conditional variance is positive, the parameters must satisfy 0, 0 and 1,2,3... .i i q = The higher order GARCH (p,q) model proposed by (Bollershev, 1986) is

generalized from ARCH (p) model by adding the moving average term as below: 2 2 2

1 1

p

t i t i t iji

q

j − −==+= + (4)

where 0, 0 0, 1,2,3. ... and 1,2,3,....i j i q j p = =

One of the extension model of GARCH (p,q) that consider the asymmetric effects observed in the data is the Threshold ARCH or TARCH model which has been introduced by (Glosten, Jagannathan, & Runkle, 1993) and (Zakoian, 1994). The simple model of TARCH (1,1) is defined as:

2 2 2 2 2

1 1 1 1t t t t tI − − − −+= + + (5)

where 11 if 0

0 otherwise

t

tI −

=

The TARCH model explains that a response of volatility is influenced by the news information either bad

or good news. The bad news 1 0t − are followed by higher volatility while the good news 1 0t − are

followed by the lower volatility. In addition, the GARCH (p,q) model are also arguably does not consider the trend and seasonality in stock prices. To capture the volatility dynamics of time varying persistence, the component GARCH model (CGARCH) has been introduced by (Engle & Lee , 1999) and (Ding & Granger, 1996) to capture the trend and seasonality in longer time period. The simplified CGARCH (1,1) equation is provided as below:

2 2 2

, ,t LR t SR t = + (6)

where



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1 1 1 2

2 2 2 2

, , 2 ( )tLR L t tt R LR LR z − − −−= + +

, 1 1

2 2 2 2

, 11 2 ( )t t tSR t SR SR SR z − − −= −+

Previous study done by (Engle, Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, 1982) assumed that the distribution was normal. Subsequently,

(Bollershev, 1986) has introduced the standardized t-distribution with degrees of freedom, ( )2v − which

given as:

( )( )( )

( ) ( )

1

221 / 2; 1

/ 2 2 2

t

v

t

v zD z v

vv v

+−

+ = +

−−

(7)

where ( )0

1vxv e x dx−

− = is the gamma function and v is the parameter that represents the thickness

of the tail distribution. If 4v , the conditional kurtosis is equal to ( ) ( )3 /2 4v v− − which above the

usual normal value of 3. For the standardized normal distribution series t , the estimation of VaR at 95%

and 99% confidence level and for one-step ahead forecast horizon is computed as:

( ) ( )5% 11 1.65 ttVaR r −= (8)

( ) ( )1% 11 2.33 ttVaR r −= (9)

For the standardized student-t distribution series 𝜀𝑡, the estimation of VaR at 95% and 99% confidence level and for one-step ahead forecast horizon is computed as:

( ) ( )5%/1% 11 v ttVaR r t = − (10)

where ( )1

2

v

v

tt

v

v

−=

−

and ( )1vt − is the correlated critical value of ( )1 − quantile , v degrees of

freedom from t-distribution. The backtesting procedure was done after VaR estimation to test the reliability and accuracy of the model used. The procedure is important to identify the amount of losses predicted by VaR model is accurate by comparing the forecasted VaR with the actual realized returns generated from the historical stock prices observation. According to (Danielsson, 2011) the VaR limit is said to have been violated given that a financial loss on a particular day exceeds the forecasted VaR as formulated below:

1 if

0 if

t t

t t

t

VaR

VaR

r

r

−

−=

(11)

The violation ratio (VR) is calculated by dividing the observed number of VaR violations and the expected

number of VaR violations. The expected number of violations is calculated as tp w where p is the



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respective probability used and tw is the number of observation used to forecast risk. The VaR model

is said to be under forecasts risk if the violation ratio is greater than one while on the other hand if it is smaller than one the VaR model is said to be over forecasts risk. To confirm whether any other value is statistically significant other than expected VR=1, the following guideline explained by (Danielsson, 2011) to explain the accuracy of the selected VaR model. The first one is if the value of the VR ∈ [0.8, 1.2] the forecast is good. If VR<0.5 or VR>1.5 the model is imprecise and if VR<0.3 or VR>2 the model is not good. Another process of backtesting involved unconditional coverage tests for the correct number of exceedances is the Proportion of Failure test (POF-test) which also known as Kupiec test. Let

1

T

ttN I

== be the number of days over a time period T when the portfolio loss was greater than the

VaR estimate given by :

, 1

, 1

1 if

0 if

t t

t

t t

t

t

rI

VaR

VaRr

+

+

=

− (12)

The unconditional coverage for 95% confidence regions introduced by (Kupiec, 1995) are presented in Table (i). The failure rate recorded follows a binomial distribution given that the number of exceptions in the sample is N and the failure rate is N/T. The no rejection region in Table (i) are defined by the appropriate log-likelihood ratio statistics:

( )( )

( )

2ln 1 2ln 1

T x xT x x

POFLx x

p p TT t

R

−−

=

− − + −

(13)

The Kupiec test is asymptotically distributed Chi-square with degree of freedom equal to one.

Table (i) Unconditional Coverage for 95% Nonrejection Region

RESULTS DATA & DESCRIPTIVE STATISTICS The data consist of approximately 3935 daily returns of FTSE Bursa Malaysia KLCI Index (KLCI) and FTSE Bursa Malaysia EMAS Shariah Index (FBMS) during trading day’s period of 2000 – 2015.

T=252 T=510 T=1000

0.01 99% N<7 1<N<11 4<N<17

0.03 97.50% 2<N<12 6<N<21 15<N<36

0.05 95% 6<N<20 16<N<36 37<N<65

0.08 92.50% 11<N<28 27<N<51 59<N<92

0.1 90% 16<N<36 38<N<65 81<N<120

p cNonrejection Region



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Table (ii) Descriptive Statistics of the daily log returns for the period 2000-2015

Both conventional and Islamic time series shows signs of volatility clustering as shown in Figure (i). The tendency of large changes in prices shows that there is persistence in the magnitude of the price changes. Based on the unconditional variance of standard deviation from Table (ii), FBMS has the highest volatility followed by KLCI. For normality test, the series must have skewness equal to zero, kurtosis coefficient equal to three and has probability of Jacque-Bera greater than 0.05 to meet the requirement of normal distribution. Table (ii) presented that KLCI and FBMS have positive skewness which showed that the returns are inclined to the right side of the distribution. The kurtosis value represents that the return distributions of KLCI and FBMS the thickness in the tail of KLCI and FBMS feature leptokurtic distribution due to coefficients larger than three. Finally, the Jacque-Bera p-value show KLCI and FBMS Index violated the normal distribution criteria.

400

800

1,200

1,600

2,000

00 02 04 06 08 10 12 14

PRICE_KLCI

4,000

6,000

8,000

10,000

12,000

14,000

00 02 04 06 08 10 12 14

PRICE_FBMS

-.12

-.08

-.04

.00

.04

.08

00 02 04 06 08 10 12 14

RETURN_KLCI

-.12

-.08

-.04

.00

.04

.08

00 02 04 06 08 10 12 14

RETURN_FBMS

Fig. (i) Continuous Compounded Daily Returns of KLCI and FBMS Index from 2000-2015

EMPIRICAL RESULTS From Figure (i), the existence of volatility clustering emerged due to the frequent changes of stock returns show unpredictability of volatility. Table (iii) shows the value of Akaike information criterion (AIC) and Schwarz information criterion (SIC) for KLCI both conventional and Islamic market for normal and student’s t distribution. Based on the information criteria of the Akaike Information Criteria (AIC) and Schwarz Information Criteria (SIC) presented in Table (iii), TARCH (1,1) is the best

Statistics KLCI FBMS

Mean -0.000144 -0.000138

Median -0.0004 -0.00038

Maximum 0.10493 0.11986

Minimum -0.04403 -0.048

Std. Dev. 0.008509 0.008977

Skewness 0.980215 1.19813

Kurtosis 14.28124 17.24644

Jarque-Bera 21496.51 34218.6

Probability 0 0

Observations 3935 3935



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estimated model for modelling the logarithmic return series of both KLCI and FBMS Index since it has the lowest values of AIC and SIC.

Table (iii)

Criteria for Model Selection

Table (iv) Parameter Estimates of the GARCH Family Model for KLCI Index (2000-2015)

AIC SIC

GARCH (1,1)-n -7.0453 -7.0373

GARCH (1,1)-t -7.1143 -7.1048

TARCH (1,1)-n -7.0541 -7.0446

TARCH (1,1)-t -7.1182 -7.1071

CGARCH (1,1)-n -7.0672 -7.0545

CGARCH (1,1)-t -7.1141 -7.0997

GARCH (1,1)-n -6.9709 -6.9629

GARCH (1,1)-t -7.0548 -7.0452

TARCH (1,1)-n -6.9808 -6.9713

TARCH (1,1)-t -7.0595 -7.0484

CGARCH (1,1)-n -6.9867 -6.9739

CGARCH (1,1)-t -7.0534 -7.0391

Index DistributionModel selection

KLCI

FBMS

normal stud t normal stud t normal stud t

C 4.00E-04 3.70E-04 2.46E-04 2.89E-04 3.75E-04 3.66E-04

AR(1) 0.1669 0.1344 0.1706 0.1371 0.1653 0.1424

1.02E-06 1.13E-06 1.17E-06 1.19E-06 4.19E-05 1.12E-04

0.1190 0.1208 0.0727 0.0825 0.0983 0.0022

0.8727 0.8687 0.8725 0.8682 0.7275 -0.9927

- - 0.0893 0.0745 0.0181 0.1226

- - - - 0.9965 0.9904

v - 5.6620 - 5.8001 - 5.7184

9.9926 17.5290 11.0080 18.1160 9.9810 17.4540

0.5310 0.0930 0.4430 0.0790 0.5321 0.0950

res squ 12.8840 12.0170 12.4290 12.1980 12.1120 11.8360

Q-(12) 0.3780 0.4440 0.4120 0.4300 0.4370 0.4590

1.0013 0.9384 0.9891 0.9780 0.9680 0.9209

0.4447 0.5069 0.4564 0.4673 0.4772 0.5247

Parameter

KLCI

GARCH (1,1) TARCH(1,1) CGARCH (1,1)

resQ-(12)

ARCH (12)



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Table (v) Parameter Estimates of the GARCH Family Model for FBMS Index (2000-2015)

Table (vi) Backtesting results using Violation Ratio for KLCI and FBMS

Table (vii) Backtesting results using Kupiec Test for KLCI and FBMS

normal stud t normal stud t normal stud t

C 4.36E-04 3.85E-04 2.65E-04 3.05E-04 4.59E-04 3.74E-04

AR(1) 0.163 0.131 0.162 0.134 0.151 0.130

9.95E-07 1.09E-06 1.07E-06 1.17E-06 4.79E-05 1.25E-04

0.127 0.112 0.076 0.074 0.058 0.012

0.869 0.879 0.871 0.876 0.785 -0.519

- - 0.094 0.080 0.023 0.114

- - - - 0.997 0.991

v - 5.066 - 5.184 - 5.079

18.349 26.039 19.449 25.662 20.429 26.494

0.074 0.006 0.054 0.007 0.040 0.005

res squ 7.931 7.276 7.316 6.423 3.597 7.258

Q-(12) 0.791 0.839 0.836 0.893 0.990 0.840

0.628 0.563 0.603 0.517 0.295 0.563

0.820 0.873 0.603 0.905 0.990 0.873

Parameter

resQ-(12)

ARCH (12)

FBMS

GARCH (1,1) TARCH(1,1) CGARCH (1,1)

95% 99%

GARCH (1,1) - n 0.8943 1.626

GARCH (1,1) - t 0.4065 0

TARCH (1,1) - n 0.813 1.626

TARCH (1,1) - t 0.4065 0

CGARCH (1,1) - n 1.3008 1.2195

CGARCH (1,1) - t 0.4065 0

GARCH (1,1) - n 1.1382 0.813

GARCH (1,1) - t 0.4878 0

TARCH (1,1) - n 0.8943 0.4065

TARCH (1,1) - t 0.3252 0

CGARCH (1,1) - n 0.9756 1.2195

CGARCH (1,1) - t 0.4878 0

Index ModelViolation ratios

KLCI

FBMS

Test Test

Stats Stats

GARCH (1,1) - n 11 -0.38 4 0.987

GARCH (1,1) - t 5 -2.136 0 -1.576

TARCH (1,1) - n 10 -0.673 4 0.987

TARCH (1,1) - t 5 -2.136 0 -1.576

CGARCH (1,1) - n 16 1.082 3 0.346

CGARCH (1,1) - t 5 -2.136 0 -1.576

GARCH (1,1) - n 14 0.497 2 -0.295

GARCH (1,1) - t 6 -1.843 0 -1.576

TARCH (1,1) - n 11 -0.38 1 -0.936

TARCH (1,1) - t 4 -2.428 0 -1.576

CGARCH (1,1) - n 12 -0.088 3 0.346

CGARCH (1,1) - t 6 -1.843 0 -1.576

KLCI

FBMS

Index Model

Binomial Test

95% 99%

N N



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Table (viii) Backtesting results using Kupiec Test for KLCI and FBMS

DISCUSSION To check the model accuracy, the backtesting procedures of violation ratio, Binomial test and Kupiec test has been done to compare the forecast VaR with the ex-post realized returns for all models. The procedure of violation ratio is occurred when financial losses exceed the previous day’s VaR. By using a sample of T = 246 days observations, the violation ratios for both conventional and Islamic indices based on 95% and 99% confidence level are considered as shown in Table (vii). The expected number of violations for 95% is calculated by using the formula explained in [6] thus, for 95% confidence level, the expected number of violation is 0.05×246=12.3 while for 99%, 0.01×246=2.46. By using 95% confidence level, the KLCI Index results exhibit that all model is said to be over forecasts risk because the value is smaller than 1 except for the CGARCH (1,1) normal distribution model. However, for student’s t distribution, all models are over forecasts risk. Different with Islamic Index (FBMS), all models over forecasts risk except for the GARCH (1,1) normal distribution model. For KLCI Index, the GARCH (1,1) and TARCH (1,1) with normal distribution displays the forecast model is good because the value of the VR ∈ [0.8, 1.2]. For FBMS Index, the TARCH (1,1) and CGARCH (1,1) normal distribution showed the forecast model is good. All model with student’s t distribution for both KLCI and FBMS Index showed that the model is imprecise because VR<0.5 or VR>1.5. Based on the Binomial test, the result show that for 5%, the critical values used are ±1.96 while for 1% the values are ±2.57. Based on 95% confidence level, the forecast VaR for all models with normal distribution are located under the non-rejection area thus the null hypothesis is accepted. However, for student’s t distribution, the forecast VaR in under rejection area which reject the null hypothesis. The result for 99% confidence level shows that all test statistic values are located under the non-rejection area thus the null hypothesis is accepted. The results of the Kupiec test are presented in Table (viii). The forecasted value of VaR for 95% confidence level with normal distribution gives good results where VaR forecast values are below than 2 3.841 = thus accepting the null hypothesis of correct

unconditional coverage. Furthermore, for student’s t distribution, the null hypothesis of correct unconditional coverage is rejected. The number of exception, N for student’s t distribution is far too small compared to the expected excesses 12.3. Additional, for 99% confidence level, the models with normal distribution yield satisfactory results with forecast VaR less than 2 6.635 = however for

Test Test

Stats Stats

GARCH (1,1) - n 11 0.1497 4 0.8188

GARCH (1,1) - t 5 5.8241 0 NA

TARCH (1,1) - n 10 0.4823 4 0.8188

TARCH (1,1) - t 5 5.8241 0 NA

CGARCH (1,1) - n 16 1.0746 3 0.1119

CGARCH (1,1) - t 5 5.8241 0 NA

GARCH (1,1) - n 14 0.2372 2 0.0928

GARCH (1,1) - t 6 4.1542 0 NA

TARCH (1,1) - n 11 0.1497 1 1.1284

TARCH (1,1) - t 4 7.9049 0 NA

CGARCH (1,1) - n 12 0.0078 3 0.1119

CGARCH (1,1) - t 6 4.1542 0 NA

KLCI

FBMS

Index Model

Unconditional Coverage

95% 99%

N N



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student’s t distribution, the result indicate NA (not applicable) due to insufficient number of convergences. CONCLUSION Based on the descriptive statistics result, both markets feature leptokurtic distribution with skewness skewed to the right and heavy tails. The TARCH (1,1) is the best fitted models based on the values of AIC and SIC. The backtesting result of violation ratio show that for 95% confidence level, the KLCI and FBMS Index show the result of over forecasts risk for both normal and student’s t. The TARCH (1,1) model with normal distribution displays as the good forecast model for both conventional and Islamic Index based on the violation value VR ∈ [0.8, 1.2]. The backtesting result of Kupiec test also show that all models with normal distribution passed the Kupiec test at 95% and 99% confidence level. However, for student’s t distribution, the result does not accept the null hypothesis of correct unconditional coverage. The Kupiec test results for student’s t distribution of both markets indicate NA (not applicable) due to insufficient number of convergences. The good forecast result for TARCH (1,1) model based on the violation ratio, Binomial test and Kupiec test are consistent with the AIC and SIC criteria to be the most accurate model to forecast VaR. Based on the given result, 95% confidence level yielded a good forecast result but when the confidence level is increased to 99%, a small sample size may lead to lack of convergence in estimating the number of exceptions. The normal distribution provides a good forecast VaR than student’s t distribution. To overcome the problem of difficulty of VaR models in estimating the heavy tailed of the return distribution, further study will be focuses on the heavy tails distribution by comparing GARCH models and the conditional Extreme Value Theory (EVT) specifications to improve the VaR forecasting performance. References Abdrashev, N. (2016). Comparison of Islamic and conventional bank stocks by Value-At-Risk

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An Estimation of Value at Risk using GARCH Models for the ...hrmars.com/hrmars_papers/An_Estimation_of_Value_at_Risk_using_GARCH_Models_for_the...conventional and Islamic stock market